Stochastic Monotonicity and Queueing Applications of Birth-Death Processes

✍ Scribed by E. A. van Doorn (auth.)

• Publisher: Springer-Verlag New York
• Year: 1981
• Tongue: English
• Leaves: 124
• Series: Lecture Notes in Statistics 4
• Edition: 1
• Category: Library

✦ Synopsis

A stochastic process {X(t): 0 S t < =} with discrete state space S c ~ is said to be stochastically increasing (decreasing) on an interval T if the probabilities Pr{X(t) > i}, i E S, are increasing (decreasing) with t on T. Stochastic monotonicity is a basic structural property for process behaviour. It gives rise to meaningful bounds for various quantities such as the moments of the process, and provides the mathematical groundwork for approximation algorithms. Obviously, stochastic monotonicity becomes a more tractable subject for analysis if the processes under consideration are such that stochastic mono tonicity on an inter­ val 0 < t < E implies stochastic monotonicity on the entire time axis. DALEY (1968) was the first to discuss a similar property in the context of discrete time Markov chains. Unfortunately, he called this property "stochastic monotonicity", it is more appropriate, however, to speak of processes with monotone transition operators. KEILSON and KESTER (1977) have demonstrated the prevalence of this phenomenon in discrete and continuous time Markov processes. They (and others) have also given a necessary and sufficient condition for a (temporally homogeneous) Markov process to have monotone transition operators. Whether or not such processes will be stochas­ tically monotone as defined above, now depends on the initial state distribution. Conditions on this distribution for stochastic mono tonicity on the entire time axis to prevail were given too by KEILSON and KESTER (1977).

✦ Table of Contents

Front Matter....Pages i-vi
Preliminaries....Pages 1-10
Natural Birth-Death Processes....Pages 11-21
Dual Birth-Death Processes....Pages 22-27
Stochastic Monotonicity: General Results....Pages 28-37
Stochastic Monotonicity: Dependence on the Initial State Distribution....Pages 38-43
The M/M/s Queue Length Process....Pages 44-65
A Queueing Model Where Potential Customers are Discouraged by Queue Length....Pages 66-71
Linear Growth Birth-Death Processes....Pages 72-75
The Mean of Birth-Death Processes....Pages 76-86
The Truncated Birth-Death Process....Pages 87-99
Back Matter....Pages 100-118

✦ Subjects

Probability Theory and Stochastic Processes